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Proc Natl Acad Sci U S A. Jul 5, 2011; 108(27): 11075–11080.
Published online Jun 17, 2011. doi:  10.1073/pnas.1018724108
PMCID: PMC3131362
Biophysics and Computational Biology

Dynamic instability-driven centering/segregating mechanism in bacteria

Abstract

All cells require the ability to process spatial information to properly position intracellular molecules. Many protein complexes and DNA molecules are actively positioned either at the cell midpoint or cell poles, but the processes which drive intracellular positioning are still poorly understood. Using computational modeling we propose a bimodal centering/segregation mechanism in bacteria which is driven by the dynamic instability of polymerizing filaments, which grow and shrink with regularity. Modeled cell centering via dynamically unstable filaments is confirmed experimentally via in vivo time-lapse, colocalization measurements of a model system of clustered plasmid-DNA centered by the dynamically unstable actin-like protein filaments Alp7A in Bacillus subtilis. Generalizing to any cylindrical cell, we find strong cell-length dependence in the centering ability of dynamically unstable filaments, culminating in pole positioning when cell length decreases significantly below the theoretically predicted average filament length. Modeling dynamic instability-driven positioning mechanisms from multiple anisotropic in vivo systems demonstrates that dynamically unstable filaments are a general mechanism for both midcell and cell-pole (segregation) positioning, and that desired positioning is preferentially selected in vivo by intrinsic filament polymerization rates and number.

Keywords: cytoskeleton, bacterial actin polymerization, mathematical modeling

In both prokaryotes and eukaryotes there are a variety of mechanisms that actively position proteins, DNA, and organelles at either the cell center or at the cell poles in preparation for cell replication and subsequent division. One mechanism for positioning material within the cell is via the generation of forces by the polymerization of dynamically unstable filaments (13). These filaments (microtubules in eukaryotes, some actin-like filaments in prokaryotes) exhibit active polymerization moderated by sudden periods of depolymerization, known as catastrophes. When an object is attached to the tail end of a dynamically unstable polymer, forces generated when that polymerizing filament comes in contact with a boundary, like a cell membrane, are often sufficient for pushing that object away from the boundary (1, 2). Combining this polymerization-driven motion with filament catastrophe allows for dynamic readjustment of the object’s position, resulting in directed motion. Microtubules, which exhibit dynamic instability, have been shown to play a role in both cell motility, and intracellular positioning via experimental and theoretical methods (38). Intracellularly, microtubules typically polymerize radially outward from their microtubule organizing centers (centrosomes), make contact with their enclosure from multiple directions, and consequently push their attached microtubule organizing center radially away from the cell membrane and toward the cell center (4, 5, 911). This microtubule-driven mechanical centering has been observed in vivo in cylindrical Schizosaccharomyces pombe cells (3, 6), alone in vitro (12), and in conjunction with motor proteins in eukaryotic cells with complex shapes (13).

In contrast, bacteria and Archaea do not contain microtubules. Instead, dynamic instability-driven positioning has been observed for two families of actin-like proteins in bacteria: ParM and Alp7A (1416), which are encoded on low copy-number plasmids in Escherichia coli and Bacillus subtilis, respectively. These filaments function to actively segregate the plasmids to the cell poles using small numbers of polymerizing filaments. Because these two filaments forming actin-like proteins are not attached to the cell membrane, and have well documented polymerization properties which are not coupled to other cell processes, they are ideal candidates for studying the positioning abilities of dynamically unstable filaments in bacteria in vivo. The presence of dynamically unstable filaments in bacteria that segregate objects raises the question of whether dynamically unstable filaments could also drive centering in bacteria. In this paper, we aim to provide understanding for how midpoint positioning and pole positioning could be achieved via dynamically unstable filaments in bacteria, and more so, to understand how these two seemingly opposite goals can be preferentially selected depending on the dynamic instability parameters of the polymers. In addition to being relevant to known dynamic instability-driven systems, this study could also be relevant to understanding systems like the ParA proteins, where evidence suggests that some can form filaments in vitro, and can function to both center and segregate plasmids under different conditions, although the mechanisms are not fully understood (17).

The best understood mechanisms for positioning molecules at the midpoint of bacterial cells are those which work on cell membranes and involve regulating the location and formation of the FtsZ cell division ring. These are the division-site-selection proteins MipZ and the Min system (18, 19). Both of these systems rely upon inhibitors that prevent the assembly of FtsZ near the cell poles. Here, we are interested in investigating whether dynamically unstable filaments can act as a mechanical positioning mechanism in prokaryotes, and in particular their possible function as a centering mechanism. We use an iterative computational model, to show how dynamically unstable filaments, in general, can drive both center and pole positioning of molecules in prokaryotes. Furthermore, by using an artificial in vivo midpoint positioning system consisting of Alp7A filaments tethered to a single DNA object in B. subtilis, we experimentally confirm the centering capabilities of a dynamically unstable filament in a bacterial cell. Finally, by comparing our computational results with known positioning of other experimental systems, including the native Alp7A segregation system, we provide predictive understanding for how dynamic instability-driven midpoint positioning can be differentiated from segregation.

Experimental Methods

Molecular Biology.

Plasmids mini-pLS20alp7A-gfp (lacO)x and mini-pLS20Δalp7AR(lacO)x were constructed as described (16) by introducing into pAID3195 and pAID3171 a fragment containing a spectinomycin resistance gene flanked by lacO arrays. These plasmids were transformed into EBS1340 (16), which expresses LacI-CFP from the xylose promoter on the chromosome of B. subtilis. Average experimental B. subtilis diameter was approximately 0.7 µm, and length was 2.9 ± 0.6 μm.

Microscopy Methods.

B. subtilis containing the mini-pLS20alp7A-gfp (lacO)x or mini-pLS20Δalp7AR(lacO)x plasmid were first grown overnight on LB-agar plates. Cells from these plates were then inoculated onto a 1.2% agarose pad containing 10% or 20% LB medium, 0.05-1% xylose and FM 4-64 (Invitrogen) at 0.2 μg/mL (20). Agarose pads on slides were incubated at 30 °C for 1–2 h and allowed to overexpress LacI-CFP, creating a single aggregated plasmid cluster per cell. Cells were imaged every 4 to 7 s on a Delta Vision Spectris Restoration Microscopy System (Applied Precision). Image acquisition speed was limited by exposure time for each of the three fluorophores and microscope mechanics to about one image per 4 s, subsequently limiting the accuracy of our time-dependent statistics. Images were deconvolved using Delta Vision SoftWoRx Image Analysis Software. Measurements were made of plasmid-focus position and filament length as a function of time and in multiple cells at one time. Image resolution was limited to 0.1 μm for length changes (~2 pixels) and 0.25 μm for absolute length. Filaments shorter than 0.25 μm were observable but not differentiable, and thus not included.

Computational Model.

A one-dimensional, stochastic, iterative numerical model of rigid dynamically unstable filaments tethered to a plasmid was used to model plasmid positioning within a uniform density bacterial cell. The model begins with either one filament growing from a randomly positioned plasmid, or two filaments growing independently from opposite sides of a plasmid. The growth of dynamically unstable filaments from the plasmid was defined by v+, the polymerization rate of the filament, v-, the rate of depolymerization, f+-, the catastrophe frequency, and f-+, the rescue frequency, or frequency of switching from catastrophe to growth (10). Filaments grew continuously in time increments of Δt = 0.1 s with rate v+ as long as the nearest monomer, monomer separation: [(cell volume)/nfree]1/3, randomly diffused to the filament tip within time Δt (time to diffuse chosen from Gaussian distribution for 1-D diffusion). Because Dprotein was estimated at 7.7 μm2/s [GFP in cytoplasm (21)], the average time for a monomer to diffuse across a cell was [double less-than sign] Δt for all cell lengths probed, and monomer concentration had minimal affect on v+. Once the elapsed time, t, reached 1/f+-, the filaments shrank with rate v- until the rescue time was reached, 1/f-+, at which point the filaments began growth again. Parameters were chosen from a Gaussian distribution constructed from the experimental mean and standard deviation. To incorporate the effects of the stall-force, which is the force necessary to stop polymerization (typically caused by a filament growing between two unmovable objects), polymerization was arrested within the simulation when the filament/plasmid complex extended the full length of the cell and filaments could no longer grow. The simulation was performed for cells of different length Lcell with the number of monomers fixed at n = 2 × Lcell/d, where d = 4 nm is the monomer size. For precise comparison of the model to our in vivo Alp7A+clustered-plasmid experimental data, we also varied the monomer concentration.

When a filament makes contact with the rigid, unmoving cell wall, monomer addition pushes the plasmid away from the wall. In the absence of filament-wall interactions, the simulated plasmid moves diffusively with Dplasmid = 8 × 10-4 μm2/s, which was experimentally measured for pLS20 plasmids (Fig. S1). For comparison, the measured plasmid diffusion coefficient in ParM systems is 4 × 10-4 μm2/s (14). Decreasing the diffusion constant, below 1 × 10-3 μm2/s does not quantitatively influence centering or segregation results from the model. Plasmid/polymer motion reaches steady state in < 200 s, and simulations are run for 500 s.

Results and Discussion

The Alp7A+Single-Plasmid-Focus System.

To experimentally evaluate the role of dynamically unstable filaments on positioning in cylindrical cells we used the Alp7A actin-like protein system encoded on the plasmid pLS20 in vivo in B. subtilis and created an artificial in vivo dynamic instability-driven centering system. A mini-pLS20 plasmid containing an Alp7A-GFP fusion and a lacO array for LacI-CFP binding enabled us to nearly simultaneously image the Alp7A polymer and the plasmid in vivo in time-lapse fluorescence microscopy. Under wild type conditions Alp7A filaments grow between two plasmids and push them toward the cell poles. Wild-type filaments have been observed to exhibit dynamic instability, including both catastrophes and rescues (16). By overexpressing LacI-CFP, the mini plasmids (N ~ 4–6) aggregated into one single cluster of plasmids per cell, appearing as a single focus of CFP fluorescence. This single focus contains approximately 60 to 90 Kb of DNA, similar in size to the native pLS20 plasmid, and is small enough to be pushed by the Alp7A polymers (Fig. 1 and Fig. 2). Growing outward from the plasmid DNA focus we observed two Alp7A filaments on opposite sides of the plasmid focus, each undergoing independent periods of growth, catastrophe, and rescue (Fig. 1, Movie S1).

Fig. 1.
(A) Time-lapse fluorescence microscopy of cells containing LacI-CFP-tagged (blue) mini-pLS20alp7A-gfp plasmids aggregated by LacI overexpression and Alp7A-GFP (green) actively polymerizing and depolymerizing outward from the plasmid (Movie S1 ...
Fig. 2.
(A and C) Time-lapse fluorescence microscopy of cells containing LacI-CFP-tagged mini-pLS20alp7A-gfp plasmids aggregated by LacI overexpression (blue), FM-464-stained membranes (red), and Alp7A-GFP (green), showing Alp7A-GFP actively moving the mini-pLS20 ...

This growth of bidirectional dynamically unstable filaments from a single focus is reminiscent of the microtubule organizing center system, only with a much smaller number of filaments and is fundamentally different from the dynamic instability observed in the native Alp7A segregation mechanisms (16). In this artificial system, two Alp7A filaments oriented toward opposite ends of the cell grew from single plasmid clusters. Multiple filaments (for example, three or more per focus) were not observed. Alp7A filaments might be bundled, but they appeared to undergo dynamic instability in unison, not independently, and with fairly uniform intensity along the filament length (Fig. S2, Movie S2). Further observation indicated that catastrophe did not require contact of the filament with the wall, nor was depolymerization complete in every instance (Fig. 1B and Fig. 2), in agreement with previous observations (16). Catastrophe may be induced in a filament when each of the two filaments are in contact with the cell wall, and one is polymerizing (for example, Fig. 2B). This confinement-driven catastrophe has been observed in microtubule systems, where filaments constrained to grow between two walls will stall their polymerization once the filament touches both sides of the confines (2). However, because of microscopy resolution limits preventing accurate assessment of the cause of catastrophe under these conditions, these events were incorporated into a universal catastrophe frequency. From time-lapse microscopy images we calculated the Alp7A filaments’ polymerization rate, v+ = 0.035 ± 0.015 μm/s, the depolymerization rate, v- = 0.069 ± 0.033 μm/s, the catastrophe frequency, f+- = 0.040 ± 0.020 s-1, and the rescue frequency, f-+ = 0.031 ± 0.025 s-1 (N =  ~ 40 growth/shrinkage cycles measured in 16 cells).

From still-frame microscopy images of multiple cells, the probability distribution of Alp7A filament lengths, P(L), as a function of filament length, L, was also measured (Fig. 1C, red circles). We find that P(L) can be fit to an exponential decay function: P(L) = const × e-L/left angle bracketlright angle bracket, where left angle bracketlright angle bracket is the average filament length, similar to measurements done in vitro for microtubules (22). The average experimental filament length derived from the exponential fit was left angle bracketlexpright angle bracket = 0.51 ± 0.06 μm (N = 329) for L > 0.25 μm. P(L) was maintained at different time points within a time lapse movie, and was maintained within one cell over time, indicating that the length distribution is in steady state. Theory predicts that for unconstrained filaments (i.e., no cell walls and a large monomer reservoir) that spend at least some time completely depolymerized (22), the average filament length depends only on the dynamic instability parameters and is equal to left angle bracketlthright angle bracket = v+v-/(v-f+- - v+f-+) (4, 5), which agrees well with length distributions of microtubules grown in vitro (22). For our Alp7A dynamic instability parameters, left angle bracketlthright angle bracket = 1.65 ± 0.85 μm, which looks qualitatively different from the experimental P(L), most likely because our Alp7A filaments were constrained both by the cell wall and by monomer concentration. We used our computational simulation to gain a better understanding of how these factors limit filament growth. When simulated filament growth was unconstrained, the average simulated filament length, left angle bracketlsimright angle bracket, was 1.47 ± 0.10 μm, which agrees with left angle bracketlthright angle bracket, confirming that our model reproduces the theoretically predicted filament length distribution (Fig. 1C, blue squares). When we modeled the growth of two filaments attached to a plasmid within the confines of a cell wall left angle bracketlsimright angle bracket = 1.12 ± 0.13 μm (Fig. 1C, downward triangles), but to obtain a similar P(L) to that experimentally observed, we also had to limit monomer concentration. Optimization of the monomer concentration to obtain the best agreement with the experimental P(L) yielded a value of left angle bracketlsimright angle bracket = 0.86 ± 0.07 μm if n = 2.9 μm/d monomers (Fig. 1C, upward triangles), which is also the maximum number of monomers to extend a filament across the cell once. The remaining discrepancies between the model P(L) and experimental P(L) suggest that there is an additional reason for the large number of small filaments observed experimentally, such as filament bundling. These results suggest that the measured length distribution of the filaments is primarily set by a combination of dynamic instability parameters, cell length, and monomer concentration.

A Single Plasmid-Focus is Pushed Away from Cell Walls by Alp7A and is Consequently Centered.

In Fig. 2, we show two examples of time lapse images (Fig. 2 Ad, Movies S3 and S4) of elongating Alp7A filaments making contact with the cell wall (labeled with stars in Fig. 2 B and D), and pushing a plasmid focus toward the cell center, showing that polymerizing Alp7A filaments generate pushing forces rather than undergoing depolymerization when contacting the cell wall. When filaments are not in contact with the cell wall, plasmid foci can move freely. The net result of the alternating filament pushing and independent plasmid motion is center positioning of a plasmid focus. Plasmid positions were determined from still-frame microscopy images and binned into 10% increments of the cell length relative to the cell center (5% on both sides of the cell center) (Fig. 3A). Centering was quantified by the fraction of plasmids in the center 20% of the cell, normalized to a random distribution (20% of the plasmids in each 20% bin), which we define as the centering factor, C = (fraction of cells in center 20%-0.2)/0.8. C = 0 for a random distribution, C = 1 is completely centered, and a negative value indicates pole positioning. In our Alp7A+plasmid-focus system, centering is apparent in the plasmid position distribution, with C = 0.31 (Fig. 3 A and B). This observed centering is in contrast to mutant cells where Alp7A is absent, in which C = -0.16. These mini-pLS20Δalp7AR plasmid foci move about at the cell poles, lacking any mechanism for directed motion (Fig. 2 E and F, Fig. S1). This migration to the edge of the cell, outside the bacterial chromosomes where the intracellular density is presumably lower, allows the large plasmids the ability to explore more free space.

Fig. 3.
(A) Fractional distribution of plasmid-focus positions in B. subtilis cells containing Alp7A polymers (open squares), and without Alp7A polymers (open circles), compared to simulated plasmid positions with two filaments with n = 2* ...

To test if this observed centering can be caused by the dynamic instability of the filaments alone, we used our computational model to simulate both the time-lapse (Fig. 3 CE) and the still-frame experiments (Fig. 3F) performed in vivo. In the absence of any filaments the simulated plasmid motion is driven solely by diffusion and is positioned at random within the cell (C = -0.01 ± 0.02, Fig. 3B), as expected for a uniform density cell, which is different from experimental cells which have a nonuniform density (Fig. 3 A and B). For experimental cell lengths, two filaments center a plasmid more accurately than one filament, with C = 0.09 ± 0.02 for one filament and C = 0.22 ± 0.02 for two filament-driven centering. Incorporating the monomer concentration limit used to best fit P(L), we find C = 0.33 ± 0.02, in agreement with the centering observed (C = 0.31) in the Alp7A+plasmid-foci system (Fig. 3B). These simulations clearly show that a sparse system of dynamically unstable filaments is sufficient to center an object within a cell, and that increasing the number of filaments from one to two helps shift the positioning of the plasmids toward the cell center. It also suggests that monomer concentration plays a role in fine-tuning the plasmid positioning, which we investigate further below.

Plasmid Positioning Dynamics Change in Cells of Different Lengths.

Qualitative observations of plasmid-foci positioning in cells of different lengths revealed a length-dependent behavior. To quantify this length dependence, still-frame fluorescence microscopy images of our Alp7A+plasmid-focus system were used to determine the number of plasmids located within the center 20% of their respective cells at one snapshot in time (Fig. 4A, red circles). For small bacteria (e.g., newly divided), the filaments interact with the cell walls frequently, pushing the plasmid focus rapidly from one side of the cell to the other in a zig-zag motion (Fig. 2 A and B). The result of this rapid pushing was that few plasmid foci were found in the center of these very short cells (Lcell < 2 μm) (Fig. 4A). As cell length increases the plasmid focus spends more time in the center of the cell, and the polymers spend less time in contact with the cell walls (Fig. 2 C and D), until the number of centered plasmids is maximized at an experimental cell length of Lcell- max = 3.6 μm, at which point C = 0.43 ± 0.11 (Fig. 4A), which is a larger value for C than that obtained by averaging C for all cell lengths (C = 0.33, average cell length is 2.9 μm). Because filaments were not found to grow longer than approximately 1.8 µm experimentally, due either to filament catastrophe or monomer concentration, cells longer than approximately 3.6 µm exhibited minimal filament-wall interactions, impeding the ability for the filaments to center any plasmids. Thus Lcell- max defines the cutoff length for maximally driven centering.

Fig. 4.
(A) Centering factor C as a function of cell length Lcell measured in Alp7A+plasmid-focus experiments (red circles), simulated with n = 2*Lcell/d (black squares) and with n = 2.9 μm/d (blue triangles). ...

To understand how dynamically unstable filaments contribute to this length-dependent centering, we turn to our iterative model. The model accurately duplicated the experimentally observed dynamic positioning including the plasmid’s zig-zag motion in short cells (Lcell [double less-than sign] Lcell- max) (Fig. 4B), and centering motion near Lcell- max (Fig. 4C). The model also qualitatively reproduced cell-length dependence of C when the monomer concentration was not fixed (n = 2 * Lcell/d) with C = 0.31 at Lcell- max = 4.5 μm, but was in near quantitative agreement with experimental results when n was optimized (n = 2.9 μm/d). In this case, the cell-length dependence of C became more peaked, with C = 0.52 ± 0.02 at Lcell- max = 3.25 μm (Fig. 4A, blue triangles). These results show that the general behavior of plasmid positioning is driven by the dynamic instability parameters alone, but that in vivo, additional constraints, like monomer concentration, fine-tune the positioning characteristics of the dynamic instability system.

Modeling the Dynamics of Plasmid Positioning: Centering vs. Pole Positioning.

Our 1-D iterative model can also probe the general positioning of any macroscopic object tethered to dynamically unstable filaments in an anisotropic cell. We modeled the cell-length dependence of the positioning of macromolecules tethered to one and two filaments for a given set of dynamic instability variables to develop some predictive understanding of the relationship between these cell-specific factors. For these simulations the monomer concentration was chosen to be independent of cell length, with n = 2 * Lcell/d, and we used the Alp7A+plasmid-foci dynamic instability variables. For both one and two filament-driven positioning, the time a tethered object spends in the center of the cell increases with cell length (Fig. 5 A and B) until Lcell- max, above which the time an object spends at the cell center decreases due to decreasing filament-wall interactions. This trend is clearly shown in the calculation of C (Fig. 5C) and the counterpart of C, which we will call the polar factor P = (fraction of cells in outer 20% of cells - 0.2)/0.8. Notice in Fig. 5C, for very short cell lengths C < 0 and P > 0, which suggests the objects are preferentially located at the cell-poles. C steadily increases and P decreases with increasing cell length. The cell length where C becomes positive and P becomes negative, Lcrossover, represents the cell length at which positioning of objects switches from pole positioning to centering. For two Alp7A filaments tethered to our plasmid foci, our simulations predict Lcrossover = 1.0 μm (Fig. 5C). Lcrossover, however, depends on the number of filaments tethered to the object, decreasing with increasing filament number (Lcrossover for one filament is 2.4 μm). Lcrossover also depends on the dynamic instability parameters of the system (see below), but did not change when monomer concentration was restricted (Fig. 4A). Above Lcrossover, two filaments are a more stable centering tool. They provide a larger range of cell lengths where plasmids are actively centered (from Lcrossover < Lcell ≤ Lcell- max), and larger values of C. Below Lcrossover, pole positioning was more predominant for single filament-driven positioning, persisting over a larger range of cell lengths (Fig. 5C). It is interesting to note that the zig-zag motion originally observed in the Alp7A-single focus experiments is reproduced both when C < 0 (Fig. 5D) and when C > 0 as long as Lcell [double less-than sign] Lcell- max (Fig. 4B), suggesting that the observed zig-zag motion is a characteristic of pole positioning, but is not necessarily an indicator of pole positioning. Summarizing these observations, our model suggests that one or two dynamically unstable filaments tethered to a macromolecule will actively center the macromolecule if Lcrossover < Lcell ≤ Lcell- max (Fig. 5E), actively pole position the macromolecule if Lcell < Lcrossover (Fig. 5D), and will have little positioning effect if Lcell[dbl greater-than sign]Lcell- max (Fig. 5F).

Fig. 5.
(A and B) Fraction of time-simulated plasmids spent at different distances from cell center for one (A) and two (B) polymers for cells of different lengths. The positions of the plasmids were binned in 5% intervals from the cell center (each 5% bin is ...

To test the robustness of these predictions across systems, we investigated three other experimental, dynamic instability-driven positioning systems: nuclear centering in S. pombe, plasmid segregation in E. coli via ParM, and plasmid segregation in B. subtilis via Alp7A using published dynamic instability parameters (Table S1). To compare dynamic instability parameters for these different systems, we used left angle bracketlthright angle bracket, as it is useful to have a single variable which combines all four parameters. left angle bracketlthright angle bracket was calculated to be 14.3 μm for Alp7A (Lcell = 2.9 μm), 6.7 μm for S. pombe (Lcell = 11.5 μm), and 3.3 μm for ParM [Lcell ~ 2 μm; for comparison, ParM in vitro has left angle bracketlexpright angle bracket = 1.5–6 μm (23, 24)]. One result which is immediately apparent is that if left angle bracketlthright angle bracket ≤ Lcell the filaments function to center their tethered macromolecules, whereas if left angle bracketlthright angle bracket > Lcell the tethered macromolecules will be predominantly pole positioned, as in segregation systems. These observations are summarized in Fig. 6D. The relationship showing that average cell length is shorter (longer) than average filament length for pole positioning (centering) is reminiscent of observations in the Min system in E. coli, where Min, a membrane tethered cell division protein, oscillates from pole-to-pole with an amplitude which is longer than the typical cell length, ensuring that pole-to-pole oscillation dominates (18).

Fig. 6.
Fractional distribution of simulated object positions of dynamic instability-driven centering (A) and segregation (B) systems after 500 s of simulated motion (N = 400). (A) Distribution of S. pombe nuclei positioned by two model ...

To test if the observed positioning could be predicted using experimentally determined dynamic instability parameters and cell length alone, we used our computational model to simulate the behavior of these three experimental systems. Our model qualitatively reproduced experimentally observed object positioning for each system (Fig. 6). For S. pombe, we were able to recreate experimentally observed centering with C = 0.56 ± 0.02 for an average cell length of 11.5 μm (Fig. 6 A and C). In our model, we simulated only two filaments, rather than the three or four filaments that are frequently used to position the nuclear material, and also incorporated the different rates of microtubule growth when the filament is pushing (0.22 μm/s) or not (0.35 μm/s) (3). The increase to three or four filaments in vivo likely further increases the centering efficacy of the dynamically unstable filaments, similar to the increase in centering from one to two filaments observed in the model (Fig. 5C).

To adapt our simulation to model the Alp7A and ParM segregation systems, we reinterpret the object-positioning in the model. Typically during segregation, a polymerizing filament grows between and pushes apart two plasmids, and then undergoes catastrophe, decoupling the plasmids, once the plasmids are sufficiently separated. This process is repeated multiple times during a single cell cycle, as filaments polymerize from the plasmids again, search for an adjacent plasmid, and push the new pair of plasmids apart once contact is made. In our model, the pole positioning of plasmids can straightforwardly be reinterpreted as a simplified measurement of the segregation of one of the plasmids in a cell with multiple plasmids. Because modeling the movement of two connected plasmid foci relative to one another requires knowledge of how the interconnecting filament will begin disassembly, the rate of which is currently unknown, we found instead that we could gain an approximate understanding of the system that agrees with the experimental data by following the position of only one of the plasmids. The positions of the other plasmids are neglected for now. For ParM dynamic instability parameters, we find that indeed the single-filament model predicts pole positioning, with P = 0.18 ± 0.02 (Fig. 6B), which is less than but comparable to results from microscopy studies (P ~ 0.38 for cells with two plasmids (25). Similarly we find that the single-filament model predicts strong pole positioning of the plasmids for the native Alp7A system with P = 0.40 ± 0.02 similar to experimental measurements (P = 0.66) (Fig. 6B). We also modeled Lcell-dependent positioning for these segregating filaments to contrast to the centering system shown in Fig. 5C. We find Lcrossover ~ 2.6 μm for ParM and 11.4 μm for native Alp7A (Fig. S4), both consistent with a prediction of pole positioning. By modeling these three additional systems we confirm that dynamically unstable filaments are not only sufficient for both center and pole positioning of objects within cells, but are also tunable to be preferentially selective for either situation.

Conclusion

We have shown, through simulations relying upon measured rates of dynamically unstable bacterial actin, that for anisotropic cells, like bacteria or fission yeast, one or two filaments can direct a tethered macromolecule to the cell center. Indeed, as few as one dynamically unstable filament can bias the position of macromolecules, like plasmid DNA, toward the cell midpoint, however symmetric filament growth of two filaments from the macromolecule results in a much higher number of centered macromolecules. Experiments in bacterial cells confirmed that bidirectional dynamically unstable filaments tethered to a plasmid focus will position it at midcell in agreement with the dynamic instability simulation. Center positioning in both Alp7A and S. pombe systems was replicated by considering only the dynamic instability parameters of the filaments and the length of the cell, even without tuning specific factors like monomer concentration, though as we show in Fig. 1C and Fig. 4A, incorporating additional constraints can significantly improve quantitative agreement to experimental data.

Our simulations also show that variations in the dynamic instability parameters, cell length, or filament numbers can rapidly change the behavior of a dynamic instability system from centering to pole positioning, allowing for the possibility of a dual centering/segregation function (Fig. 6D). These results suggest a robust versatility of dynamically unstable filaments, which are built into so many biological systems. For example, we have experimentally shown that a single focus can be centered over a wide range of cell lengths using bidirectional Alp7A filaments, in direct contrast to segregation observed previously when Alp7A filaments are located between plasmids, as found in cells with a higher number of plasmids (16). In addition the modeled cell-length-dependent positioning could be useful for cellular mechanisms that require dynamic readjustment during the cell cycle, such as bacterial origins of replication, regulatory proteins, or cell division complexes (26). This dual centering/segregation function is also seen in the plasmid partitioning mechanism of the ParA family, which exhibits both centering of a single plasmid and segregation of plasmids for larger numbers of plasmids, and though the mechanism may not necessarily be the same, it points to a more general need for multifunctional positioning mechanisms within cells (2729), which could be filled by dynamically unstable filaments, or other force generating mechanisms. From a different perspective, the relationships quantified with the computational model allow for a predictive capability, which might be useful for synthetic biology, where dynamically unstable filaments could be tuned to play the necessary positioning role within a synthetic cell.

Supplementary Material

Supporting Information:

Acknowledgments.

Thanks to E. Becker and A. Derman for helpful discussions and Alp7 plasmids, K.C. Huang, J. Ptacin, and W. Margolin for careful reading of the manuscript. This work was supported by National Institutes of Health Grants R01GM073898 and R01GM084334 (to J.P.).

Footnotes

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1018724108/-/DCSupplemental.

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